A typical problem involving the angles formed by secants and tangents in a circle gives us information about the measures of the angle exterior to the circle and/or about the measures of the intercepted arcs of the circle. Two examples of this type of problem are presented below.

In the diagram shown below, circle O has arcs CB and CD with measure x + 10 and 5x – 20 respectively. What is the measure of each arc if angle P has measure x° ?

What is your answer?

Examples

Two secants in circle O intersect in point P outside the circle. These secants intercept arcs of length x + 7 and 3x - 9 respectively. If angle P has a measure of 20°, what is the measure of the smaller intercepted arc?

25°

35°

75°

28°

What is your answer?

A secant and a tangent form an angle A outside circle O. The intercepted arcs have measures of 190° and x + 20. If angle A has a measure of 50°, what is the measure of the arc of the circlenot intercepted by the secant and tangent?

120°

80°

30°

What is your answer?

When an angle is formed outside a circle either by two secants or one secant and a tangent, there is a relationship between the measure of this angle and the difference of the measures of the intercepted arcs.

We can use this equation to relate measures and in some cases, find a value of x if one or more of the measures is given as an expression in terms of x. In solving the equation, it is important to subtract carefully. The solution process can then be simplified by multiplying both sides of the equation by 2.